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\begin{enumerate}
  \item Verify that the proposition $p\lor \lnot (p\land q)$ is a tautology.

  \item Show that the proposition $\lnot(p \land q)$ and $\lnot p\lor \lnot q$ are logically equivalent.

  \item Prove that $\lnot (p \land q)\lor (\lnot q \land q)\equiv \lnot p$.

  \item Consider the conditional proposition $p\to q$. The simple propositions $q\to p, \lnot p \to \lnot q$
  and $\lnot q \to \lnot p$ are called, respectively, the \textsl{converse}, \textsf{inverse},
  and \textsl{contrapositive} of the conditional $p \to q$. Which of these are logically equivalent to $p \to q$?

  \item Show that the following argument is a fallacy: $p \to q$, $\lnot p \vdash \lnot q$.

  \item Determine the validity of the following argument: $p \to q, \lnot p \vdash \lnot q$.

  \item Prove the following argument is valid: $p \to \lnot q, r \to q, r\vdash \lnot p$.

  \item Negate each of the following statements:

  i. $\exists x\ \forall y, p(x, y)$;
  ii. $\forall x\ \forall y, p(x, y)$;
  iii. $\exists x\ \exists y\ \exists z, p(x, y, z)$.

  \item Find the truth tables for: i. $p\lor \lnot q$; ii. $\lnot p \land \lnot q$.
  \item Verify that the proposition $(p\land q)\land \lnot (p \lor q)$ is a contradition.
\end{enumerate}

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